1
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2answers
63 views

Worst case examples of non-differentiability of the Riemannian distance function

Let $g$ be a $C^\infty$ Riemannian metric on the plane, and let $p$ be a point on the plane. Let $X$ be the set of points $x$ at which the Riemannian distance $d(p,x)$ is not differentiable. How bad ...
1
vote
1answer
109 views

Show that the Lie derivative is equal to the commutator

Let $\Omega \subseteq \mathbb{R}^d $ be open. Let $\epsilon > 0$. Let $(\phi_t)_{t \in (-\epsilon , \epsilon)} $ be a family in $\mathrm{Diff}(\Omega)$ such that $ \phi_0 = id_{\Omega}$ and ...
3
votes
1answer
214 views

Exponential map on manifolds and differential

I am trying to understand the proof of Theorem 3.7, page 72 of Riemannian Geometry by M. Do Carmo. For $M$ a Riemannian manifold and $(U,\varphi)$ a chart around a point $p\in M$, he (more or ...
1
vote
1answer
318 views

Commutators, and Christoffel symbols in a non holonomic basis

I have a frame that varies along a curve $\gamma$ : the frame consists in the tangent vector of the curve plus some constant non orthogonal vectors. I need to compute ...
3
votes
1answer
145 views

Derivative of a parallel translation inside a metric

Let $M$ be a riemannian manifold with metric $g$ and a connection $\nabla$ on $M$. Let $X,Y$ two vector fields along a curve $\gamma$ on $M$. Let $$\tau_{t,s}:T_{\gamma(s)}M\to ...
2
votes
1answer
166 views

Derivative of a metric tensor along a curve

Let $M$ be a Riemannian manifold with metric tensor $g$ and Levi-Civita connection $\nabla$. Also, let $u: \mathbb{R}\to T_pM$ be a smooth curve in $T_pM$. In a proof, my course notes assure that ...